Theorem finlocfin 23442

Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypotheses

Ref	Expression
finlocfin.1	⊢ 𝑋 = ∪ 𝐽
finlocfin.2	⊢ 𝑌 = ∪ 𝐴

Assertion

Ref	Expression
finlocfin	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Proof of Theorem finlocfin

Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1133	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐽 ∈ Top)
2		simp3 1135	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
3		simpl1 1188	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ Top)
4		finlocfin.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5	4	topopn 22826	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6	3, 5	syl 17	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ 𝐽)
7		simpr 483	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8		simpl2 1189	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ Fin)
9		ssrab2 4069	. . . . 5 ⊢ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴
10		ssfi 9196	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)
11	8, 9, 10	sylancl 584	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)
12		eleq2 2814	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑥 ∈ 𝑛 ↔ 𝑥 ∈ 𝑋))
13		ineq2 4200	. . . . . . . . 9 ⊢ (𝑛 = 𝑋 → (𝑠 ∩ 𝑛) = (𝑠 ∩ 𝑋))
14	13	neeq1d 2990	. . . . . . . 8 ⊢ (𝑛 = 𝑋 → ((𝑠 ∩ 𝑛) ≠ ∅ ↔ (𝑠 ∩ 𝑋) ≠ ∅))
15	14	rabbidv 3427	. . . . . . 7 ⊢ (𝑛 = 𝑋 → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅})
16	15	eleq1d 2810	. . . . . 6 ⊢ (𝑛 = 𝑋 → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin))
17	12, 16	anbi12d 630	. . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)))
18	17	rspcev 3601	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
19	6, 7, 11, 18	syl12anc 835	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
20	19	ralrimiva 3136	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
21		finlocfin.2	. . 3 ⊢ 𝑌 = ∪ 𝐴
22	4, 21	islocfin 23439	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
23	1, 2, 20, 22	syl3anbrc 1340	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  {crab 3419   ∩ cin 3938   ⊆ wss 3939  ∅c0 4318  ∪ cuni 4903  ‘cfv 6543  Fincfn 8962  Topctop 22813  LocFinclocfin 23426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7869  df-1o 8485  df-en 8963  df-fin 8966  df-top 22814  df-locfin 23429

This theorem is referenced by:  locfincmp  23448  cmppcmp  33516